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Magazine Chapter 30: Problem 80
What is the ratio of the kinetic energies for an alpha particle and a beta particle if both make tracks with the same radius of curvature in a magnetic field, oriented perpendicular to the paths of the particles?
Short Answer
Expert verified
The kinetic energy ratio is approximately \(5.45 \times 10^{-4}\).
Step by step solution
01
Understanding the Given Problem
We are asked to find the ratio of the kinetic energies for an alpha particle and a beta particle given they both follow paths with the same radius of curvature in a magnetic field. Both particles move perpendicular to the magnetic field.
02
Apply the Formula for the Radius of Curvature
The radius of curvature \( r \) in a magnetic field \( B \) is given by the formula \( r = \frac{mv}{qB} \), where \( m \) is the mass of the particle, \( v \) is its velocity, and \( q \) is its charge. For both particles, \( r \) is the same.
03
Relate Radius to Kinetic Energy
Kinetic energy \( KE \) is given by \( KE = \frac{1}{2}mv^2 \). We can use the expression for radius to relate \( mv \) to \( r, q, \text{ and } B \). Substituting for \( mv \) from the formula \( mv = qBr \), we get \( v = \frac{qBr}{m} \).
04
Express Kinetic Energy in Terms of Known Quantities
Substitute \( v = \frac{qBr}{m} \) into the kinetic energy formula: \( KE = \frac{1}{2}m\left(\frac{qBr}{m}\right)^2 = \frac{1}{2}\frac{(qBr)^2}{m} = \frac{q^2B^2r^2}{2m} \).
05
Compare the Kinetic Energies
We seek the ratio of kinetic energies of the alpha particle \( KE_{\alpha} \) and beta particle \( KE_{\beta} \). This is given by \( \frac{KE_{\alpha}}{KE_{\beta}} = \frac{\frac{q_{\alpha}^2B^2r^2}{2m_{\alpha}}}{\frac{q_{\beta}^2B^2r^2}{2m_{\beta}}} \). The \( B^2r^2 \) terms cancel, leaving \( \frac{KE_{\alpha}}{KE_{\beta}} = \frac{q_{\alpha}^2 / m_{\alpha}}{q_{\beta}^2 / m_{\beta}} \).
06
Substitute Values for Charge and Mass
An alpha particle has a charge \( q_{\alpha} = 2e \) and a mass \( m_{\alpha} = 4m_n \), where \( m_n \) is the nucleon mass. A beta particle has a charge \( q_{\beta} = e \) and a mass \( m_{\beta} \) which is the electron mass \( m_e \). Substitute these values into the ratio: \( \frac{(2e)^2 / 4m_n}{e^2 / m_e} = \frac{4e^2 / 4m_n}{e^2 / m_e} = \frac{m_e}{m_n} \).
07
Calculate the Final Ratio
Substituting the known mass values \( m_n = 1.67 \times 10^{-27} \text{ kg} \) and \( m_e = 9.11 \times 10^{-31} \text{ kg} \), compute \( \frac{m_e}{m_n} \approx \frac{9.11 \times 10^{-31}}{1.67 \times 10^{-27}} \approx 5.45 \times 10^{-4} \). Therefore, \( \frac{KE_{\alpha}}{KE_{\beta}} \approx 5.45 \times 10^{-4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Alpha Particle
An alpha particle is a type of ionizing radiation ejected by certain radioactive materials. It consists of two protons and two neutrons bound together, which is the same as a helium nucleus. This composition results in an alpha particle being relatively massive and positively charged with a charge of \( +2e \).
- **Mass**: About four atomic mass units (amu), which corresponds to roughly \( 4m_n \), where \( m_n \) is the nucleon mass.
- **Charge**: Twice the elementary charge \( e \), i.e., \( 2e \).
Beta Particle
Beta particles are high-energy, high-speed electrons or positrons emitted during the radioactive decay process of an unstable atomic nucleus. They have significantly less mass compared to alpha particles.
- **Mass**: Essentially that of an electron, \( m_e \), which is about \( 1/1836 \) the mass of a proton.
- **Charge**: Typically the elementary charge \( e \), but negatively charged if they are electrons and positively if they are positrons.
Magnetic Field
A magnetic field is a field that exerts magnetic force on moving electric charges, causing them to move along a circular path when crossing the field perpendicularly. The strength and direction of this force are dictated by the Lorentz force law.
- **Force**: Calculated as \( F = qvB \), where \( q \) is the charge, \( v \) the velocity, and \( B \) the magnetic field strength.
Radius of Curvature
The radius of curvature for a charged particle moving in a magnetic field is an important parameter that helps analyze the particle's motion. It is derived from the balance between magnetic force and centripetal force.
- **Formula**: The formula \( r = \frac{mv}{qB} \) relates radius \( r \) to particle mass \( m \), velocity \( v \), charge \( q \), and magnetic field strength \( B \).
